nLab stratified simplicial set

Idea

A stratified simplicial set is a simplicial set equipped with information about which of its simplices are to be regarded as being thin in that they are like identities or at least like equivalences in a higher category.

The theory of simplicial weak ∞-categories is based on stratified simplicial sets.

Definition

A stratification of a simplicial set $X : \Delta^{op} \to Set$ is a subset $t X \subset \coprod_{[n]} X_n$ of its set of simplices (not in general a simplicial subset!) such that

• no 0-simplex of $X$ is in $t X$;

• every degenerate simplex in $X$ is in $t X$.

A stratified simplicial set is a pair $(X, t X)$ consisting of a simplicial set $X$ and a stratification $t X$ of $X$.

The elements of $t X$ are called the thin simplices of $X$.

For $(X, t X)$ and $(Y, t Y)$ stratified simplicial sets, a morphism $f : X \to Y$ of simplicial sets is set to be a stratified map if it respects thin cells in that

$$f(t X ) \subset t Y \,.$$

The category of stratified simplicial sets and stratified maps between them is usually denoted $Strat$.

This category is a quasitopos. Hence, in particular, it is cartesian closed.

Examples

• Every simplicial set gives rise to a stratified simplicial set

  • using the maximal stratification: all simplices of dimension >0 are regarded as thin;

  • using the minimal stratification: only degenerate simplices are thin.

  These two stratifications give left and right adjoints to the forgetful functor from stratified simplicial sets to simplicial sets.

• The standard thin $n$-simplex is obtained from $\Delta[n]$ by making its only non-degenerate $n$-simplex thin.

• The $k$th standard admissible $n$-simplex $\Delta^a_k[n]$, defined for $n \geq 2$, $0 \lt k \lt n$, is obtained from $\Delta[n]$ by making all simplices $\alpha \colon [m] \to [n]$ with $k-1,k,k+1 \in$ im$(\alpha)$ thin.

• The standard admissible $(n-1)$-dimensional $k$-horn $\Lambda^a_k[n]$, defined for $n \geq 2$, $0 \lt k \lt n$, is the pullback of the stratified simplicial set $\Delta^a_k[n]$.

• A complicial set is a stratified simplicial set satisfying certain extra conditions. Complicial sets are precisely those simplicial sets which arise (up to isomorphism) as the ∞-nerve $N(C)$ of a strict ∞-category $C$, where the thin cells are the images of the identity cells of $C$.

• A simplicial set is a Kan complex precisely if its maximal stratification makes it a weak complicial set.

The category of stratified simplicial sets

There are several tensor products on the category $Strat$ of stratified simplicial sets that make it a monoidal category.

Strat with the Verity-Gray tensor product

Consider the monoidal category $(Strat, \otimes)$ where $\otimes$ is the Verity-Gray tensor product.

(Notice that this is not closed, as far as I understand.)

Using the canonical stratification of ∞-nerves on strict ∞-categories as complicial sets, the $\omega$-nerve is a functor

$$N : Str \omega Cat \to Strat \,.$$

Proposition

The functor $N : Str \omega Cat \to Strat$ has a left adjoint $F : Strat \to Str \omega Cat$ which is a strong monoidal functor.

Or so it is claimed on slide 60 of Ver07

References

A useful quick introduction is the beginning of these slides:

• Dominic Verity, Weak Complicial sets and internal quasi-categories (pdf slides)